Problem Solving or the ART of THINKING ON IT “mathematical thinking can be improved by tackling questions conscientiously; reflecting on this experience; linking feelings with action; studying the process of resolving problems; and

The subject of Geometry is best understood by those who are willing to learn and understand definitions,theorems and postulates and who can apply this knowledge to problem solving. Problem solving may be

in theform of a formal two-column proof or a paragraph proof or in the form of developing an equation and solving.

There are some strategies

to problem solving:(1) being stuck is good as it generates thinking skills, (2) focused“thinking on it” is astart to becoming unstuck, (3) specializing or trying some specific cases leads to possiblesolutions and/or ideas, (4) try to generalize after specializing and noting a pattern, (5) jot down ideas as theyflow from the brain, (6) write down everything that you know about the problem, (7) ask yourself WHY,

(8) draw pictures and/or re-arrange the information, (9) reduce the problem to a smaller problem, (10) walkaway from the problem for a period of time to clear the mind.

1.

How many squares are there on a standard chessboard?

2.

Draw a square on your paper. Draw a line across it. Draw several more lines through the square so that

several regions are formed. The task is to color regions in such a way that adjacent regions are nevercolored the same. How few different colors are needed to color any arrangement?

3.

A goat is tethered by a 6 meter rope to the outside corner of a shed measuring 4 meters by 5 meters inthe middle of a pasture. What is the area of grass on which the goat can graze?

4.

Five women have lunch together seated around a circular table. Ms. Osborne is sitting between Ms.Lewis and Ms. Martin. Ellen is sitting between Cathy and Ms. Norris. Ms Lewis is between Ellen andAlice. Cathy and Doris are sisters. Betty is seated with Ms. Parkes onher left and Ms. Martin on herright. Match the surnames with the first names.

5.

Ross collects lizards, beetles and worms. He has more worms than lizards and beetles together. Alltogether there are twelve heads and twenty-six legs. How many lizards are

there in the collection?

6.

Small booklets can be made by folding a single sheet of paper several times and then cutting andstapling. I would like to number the pages before making the folds. How can this be done? (considerbooklets of various number of

pages)

7.

A secret number is assigned to each vertex of a triangle. On each side of the triangle is written the sumof the secret numbers at the endpoints. Give a rule for finding the secret numbers if you know the sums.

8.

Twenty-five coins are placed in a 5 by 5 array. A fly lands on one coin and tries to hop onto every coin

exactly once by only moving to an adjacent coin in the same row or column. Is it possible?

9.

Take a strip of paper and fold it in half several times. Unfold it and notice how many folds are “in” and

how many folds are “out”. What sequence would arise from 8 folds?, 9 folds?, 10 folds?

10.

Some numbers can be written as the sum of consecutive numbers. For example: 9 = 4 + 5

11 = 5 + 6

21 = 6 + 7 + 8

Are there any numbers that cannot be written as the sum of consecutive numbers?